Exploring the Optimal Replication Strategy in P2P-VoD Systems: Characterization and Evaluation

Transcription

1 This paper was presented as part of the main technical program at IEEE INFOCOM 211 Exploring the Optimal Replication Strategy in P2P-VoD Systems: Characterization and Evaluation Weijie Wu John C.S. Lui Computer Science & Engineering Department The Chinese University of Hong Kong Abstract Content providers of P2P-Video-on-Demand (P2P- VoD) services aim to provide a high quality, scalable service to users, and at the same time, operate the system with a manageable operating cost. Given the volume-based charging model by ISPs, it is to the best interest of the P2P-VoD content providers to reduce peers access to the content server so as to reduce the operating cost. In this paper, we address an important open problem: what is the optimal replication ratio in a P2P- VoD system such that peers will receive service from each other and at the same time, reduce the traffic to the content server. We address two fundamental problems: (1) what is the optimal replication ratio of a movie given its popularity, and (2) how to achieve the optimal ratios in a distributed and dynamic fashion. We formally show how movie popularities can impact server s workload, and formulate the video replication as an optimization problem. We show that the conventional wisdom of using the proportional replication strategy is non-optimal, and expand the design space to both passive replacement policy and active push policy to achieve the optimal replication ratios. We consider practical implementation issues, evaluate the performance of P2P-VoD systems and show that our algorithms can greatly reduce server s workload and improve streaming quality. I. INTRODUCTION Adapting the peer-to-peer (P2P) technology into video-ondemand (VoD) service has received a lot of attention in the past few years. There are a number of successful P2P-VoD systems like PPLive, PPStream and UUSee. The key advantage of using this technology is that the system can utilize peers resources to satisfy other peers viewing requirement, and thereby achieve a more scalable system as compared to the traditional client-server architecture. The main difference between P2P-VoD systems and P2P live streaming systems is that peers in P2P-VoD service need to contribute a much larger local cache space 1 as well as their upload bandwidth. For example, in PPLive system [5], each peer needs to dedicate 1 GB of local storage for a scalable P2P-VoD service. This local storage is for caching some previously watched movies so that in case a new peer arrives, this peer does not need to download from the VoD server but rather, obtains the data directly from other peers. This unique feature not only frees up the server to serve other peers, but more importantly, reduces the upload bandwidth traffic of P2P-VoD content providers. Since content providers usually pay ISPs based on the volumebased charging scheme, it is to the best interest of a P2P- 1 In this paper, local cache space refers to physical memory or storage space that is dedicated to cache any multimedia data. VoD content provider to reduce the server s upload traffic, and a proper caching policy can achieve such goal. Given the distributed cache spaces of all peers, we aim to address how to utilize them efficiently so as to reduce the operating cost of the P2P-VoD content provider. Specifically, given the caching and upload capacities, what is the optimal caching policy so as to minimize the traffic to the P2P-VoD server? Unlike the P2P file sharing applications, to support peers in a P2P-VoD service, one has to guarantee (a) a low start-up latency in accessing the movies; (b) supporting the required playback-rate for all peers which want to watch the movie. Therefore, a good replication policy should duplicate enough replicas so as to support the required workload. To illustrate, consider a P2P-VoD system with 1 cache units (i.e., each cache unit can store one movie). There are two movies M 1,M 2 with the same viewing popularity, i.e., each with 5 viewers. Assume a replication policy keeps 99 replicas for M 1 but only one replica for M 2. Then, requests for M 2 will fail to find enough peers support and have to go to the server, yet there are excessive replicas of M 1 and some peers which cache M 1 will not be using their upload bandwidth efficiently. Characterizing the optimal caching policy is challenging, especially when peers have different upload capacities and movies have different viewing popularities. These factors make designing the optimal replication policy an open problem so far. In this paper, we address the following problems: Given movies viewing popularities, what is the optimal ratio of total storage space that should be dedicated to each movie such that when a new peer arrives, it is most likely to be served by other peers in the system? Given the system dynamics (e.g., peers churn), how to design distributed and adaptive algorithms to achieve the optimal replication ratios? Previous work [5] and conventional wisdom suggested to use the proportional replication strategy, i.e., replicating movies proportionally to their popularities. However, authors in [13] reported a contradicting result that the proportional replication may have poor performance, especially for unpopular movies. In this work, we give the formal justifications on the above contradiction and provide insights to characterize the optimal replication ratios. These insights provide us with the principles in designing practical and effective replication algorithms. The contributions of our paper are: /11/$ IEEE 126

2 We show that proportional replication is not the optimal strategy. Instead, one should replicate movies proportionally to the deficit bandwidth (we will formally define this concept in Sec. II-B). The optimal replication strategy should be more greedy for unpopular movies. We expand the design space by considering both passive and active replication. We propose a passive replacement algorithm to decide which movie to purge when a local storage is full; we also propose an active replication algorithm to aggressively push data to peers so as to achieve the desired replication ratios. We evaluate our proposed algorithms and show significant improvement of video quality and large reduction on server s workload. This is the outline of our paper. In Section II we present a mathematical model for P2P-VoD system and give an optimization framework for the replication problem. In Section III, we discuss the optimal replication ratios in software based systems. In Section IV, we present our passive replacement and active push algorithm to achieve the optimal replication ratios. In Section V, we consider several important practical issues in implementation and present the performance results of our algorithms. Section VI states the related work and Section VII concludes. II. MATHEMATICAL MODEL Let M = {M 1,...,M K } be the set of movies which are stored in S, the logical server of the P2P-VoD system 2. Peers which want to watch movie M k can access the data from (1) S, or (2) any other peer which is watching M k, or (3) peers which have watched and cached M k. A P2P-VoD content provider aims to minimize, or at least to reduce, the upload bandwidth traffic of S so as to reduce its operating cost. The upload consumption of S is affected by (1) the peer scheduling policies, and (2) movie replication strategies. For the ease of presentation, we decompose these two design problems and use a realistic and practical peer scheduling strategy: a peer first requests data from other peers, and only when other peers cannot supply sufficient bandwidth, this peer requests data from S. In here, we focus on the movie replication strategy and show how it can significantly impact the upload bandwidth consumption of S. In what follows, we first state the general P2P-VoD setting, then we give a general model to show how replication strategy can affect the upload bandwidth consumption of S. A. Peer Type and Peer Scheduling Policy In general, there are two types of peers: Active peers: an active peer is one which is currently watching a movie and at the same time, provides upload service to other peers. The data that this peer uploads may come from the movie it is currently watching, or the data it has stored in its local cache. Inactive peers: an inactive peer is one which is currently not watching any movie, but it stays online in a P2P 2 A logical server may consist of many physical servers which process any request from peers in a P2P-VoD system. system, and this peer contributes its upload bandwidth to other peers. This type of peers can represent those online set-top boxes which have built-in P2P functionalities. We differentiate these two types of peers because their distribution in a P2P-VoD system will have a major impact on replication strategies. A peer scheduling strategy includes two design issues: (1) from a downloader s viewpoint, it needs to decide which peers to request the data from; (2) from an uploader s viewpoint, it needs to decide which peers to upload the data to when it receives multiple requests. Let us first describe how an active peer downloads a movie. Suppose a peer wants to watch movie M k. In general, its request can be supported by three different sources: (a) the server S; (b) active peers which are currently watching the same movie M k, and we denote these as the concurrent peers; (c) peers which are not watching movie M k but have stored M k at their local cache, and we denote these as the replication peers. Note that a replication peer may be an active peer watching another movie; or an inactive peer which has movie M k in its cache. Lastly, in order to have a short start-up latency and good streaming quality, each peer needs to download the movie at its playback rate r. We consider the following practical peer scheduling policy: a peer first seeks help from concurrent peers; if the playback rate cannot be satisfied, the peer seeks help from replication peers; if the playback rate still cannot be satisfied, the server S will upload the data to this peer. We would like to comment on the rationale of this scheduling policy. We let S be the last requesting target of a peer in order to reduce the upload consumption of S so as to reduce the operating cost of the P2P-VoD system. Note that, in real systems, a peer may cache only part but not the whole movie (details are discussed in Section V). Therefore, requesting from a replication peer may result in frequent rescheduling: a peer P i that requests data from a replication peer P j needs to look up other peers for support when P i s viewing part is not cached in P j ; however, it is more likely that P i can download from one particular concurrent peer P k throughout the watching period. This is why we prefer concurrent peers over replication peers to support the viewing requirement. One interesting question is how should a peer seek help from concurrent peers? Note that only those concurrent peers which arrived earlier than this peer can assist in data upload. We use the sequential-search-for-help principle: suppose there are n k concurrent peers watching movie M k, and these concurrent peers are denoted as P 1, P 2,..., P nk, with P 1 having the earliest arrival time. In here, P 1 cannot get any download from other concurrent peers but instead, may ask replication peers or S for help, while the only concurrent peer that P 2 can seek service from is P 1. In general, peer P i will first seek help from P i 1 ; if the request cannot be satisfied, P i will send request to peers P i 2, P i 3,..., in a sequential manner, until it receives data to satisfy its playback rate r. From an uploader s viewpoint, it may receive multiple download requests simultaneously. In our work, an active 127

3 peer will serve its concurrent peers with a higher priority. Only when its upload bandwidth is not fully consumed by concurrent peers, the active peer will provide the remaining upload bandwidth to other peers. For an inactive peer, it will upload to those peers watching any movie that is stored in its local cache. The assumption is not only for mathematical tractability, but this simple scheduling rule can also maximize the number of peers to receive data at the playback rate r. B. Model of Movie Popularity on Server s Workload Let us illustrate how the movie popularity can affect the server s upload bandwidth. First, peer P i s download rate can be supported partly by concurrent peers, partly by replication peers, and partly by the server S. Let d c i and d r i be the total download rates supported by concurrent peers and replication peers, respectively. If d c i + d r i is below the playback rate r, the server needs to fill up the gap r d c i d r i so as to satisfy the QoS guarantee. For the ease of presentation, we denote d s i = r d c i d r i ( i N A) as the filling upload rate from the server to P i, where N A is the set of all active peers. Denote N k as the set of active peers which are currently watching movie M k. Obviously, we have N k N A. The expected upload bandwidth consumption of the server S is denoted as U, which can be expressed as: U = E d s i = E d s i. (1) i N A i N k k M Based on the peer scheduling policy above, we have: Lemma 1: In a large scale P2P-VoD system, the expected upload consumption at the server is approximately U + E (r d c i) R k, (2) i N k k M where R k is the maximal upload bandwidth from all replication peers that can contribute to M k, or i N k d r i R k; and (x) + =max(,x). Due to the page limit, we omit the proofs of lemmas and propositions in this paper. Interested readers may refer to our technical report [16] for details. Let us define the following important notation. Definition 1: For movie M k, we define deficit bandwidth as D k (n k )= (r d c i)= ( d r i + d s i ), (3) i N k i N k and let D k (n k )=E Dk (n k ) be the expected deficit bandwidth of having n k concurrent peers in N k, i.e., the bandwidth gap between the total required playback rates and the total download rates that can be supported by n k concurrent peers. We can now rewrite Eq. (2) and directly obtain the following lemma: Lemma 2: Given the number of concurrent peers in each movie, the expected upload consumption at the server is U = (D k (n k ) R k ) +. (5) k M (4) Our objective is to find a replication strategy, which determines R k, such that U, the average upload bandwidth consumption of the server S, can be minimized. Thus, we want to explore how the popularity of a movie may affect D k (n k ), which in turn will affect the choice of replication. In what follows, we state an important lemma: Lemma 3: Deficit bandwidth can be iteratively calculated by D k (n k )=n k r d c i, (6) i N k, d c i = 1; i = i 1 min j=1 (u j d c j ),r, i 2, where u j denotes the maximal upload rate of P j. Let us use an example to illustrate the above framework in calculating d c i and D k (n k ). Assume six peers are watching movie M k. These peers are P 1,...,P 6, with P 1 arriving the earliest. We fix the movie playback rate at r = 5 kbps. If we know their upload capacities u i as illustrated in Tab. I, we can calculate d c 1,..., d c 6 respectively, and by summing up all (r d c i ) we can get the deficit bandwidth, or D k (6) = 1 kbps. This D k (6) represents the bandwidth required to support all these six peers. P 1 P 2 P 3 P 4 P 5 P 6 u i (kbps) d c i (kbps) D k (6) (kbps) 1 TABLE I: Example for calculating d k i and D k (n k ) In summary, given each peer s upload capacity (or u i ), one can determine d c i and D k (n k ). Furthermore, given the upload capacity distribution of N k, we may determine D k (n k ) by taking the expectation. Based on this framework, we can show how movie s popularity may impact the upload bandwidth consumption of the server S. In Section III, we will show some interesting implications. C. Model of Replication on Server s Workload Let us show how replication can help to reduce the server s workload. Intuitively, without using replication (or when d r i = ), D k (n k ) will be the upload bandwidth consumption of server S for movie M k. Replication helps to reduce server s upload bandwidth consumption since peers could use replicas to partially support the deficit bandwidth D k (n k ). For the ease of presentation, we have the following assumptions: The system contains K movies and N peers, of which αn are active peers. An active peer watches movie M k {M 1,..., M k } with probability ρ k, with K k=1 ρ k =1. Each peer, either active or inactive, caches one movie that it has watched before. An active peer can also upload the movie it is currently watching to other peers 3. 3 Note that the typical cache size of a peer is 1 GB (e.g., in PPLive), and typical movie size is about 5 MB. Therefore, we assume that besides the currently watching movie, an active peer can cache one extra movie so as to cope with typical settings of real systems. (7) 128

4 Peers scheduling strategy follows the assumptions in Section II-A. To model the distribution of number of active peers in each movie, one can use a closed queueing network to represent the P2P-VoD system. Let random variable n k denote the number of peers watching movie M k. Similar to the work in [14], we can express the probability for movie M k having n k viewers, for k =1, 2,...,K, as: q k P (n 1 = n 1,...,n K = n K )=(αn)! ρn1 1 K n 1!...ρnK n K!, (8) with K k=1 n k = αn. Here, ρ i implies the relative popularity of movie M i. Based on this queueing model, we have the following proposition: Proposition 1: The average upload consumption of S is U(q) = (αn)! ρn1 1 K n 1!...ρnK n K! K D k (n k ) nk =αn k=1 K + (Cl u (n l )+D l (n l ) n l r) q k CI u ((1 α)n), (9) l=1 where q k is the fraction of cache space dedicated to M k, C u k (n k)=e i N k u i, C u I (n I )=E i N I u i, and N I is the set of inactive peers. Now the model of the optimal replication strategy can be formulated as follows. We seek to find a set of replication ratios q which satisfies: min q U(q) subject to q T 1 =1. (1) In other words, we want to determine the number of replicas for each movie M k M, so that the average upload bandwidth consumption of S can be minimized. In general it is difficult to get a close form solution for this optimization problem. However, as we will show in the following section, when we consider some common deployment cases, one can get some interesting implications and insights in designing replication strategies. III. OPTIMAL REPLICATION UNDER SOFTWARE DEPLOYMENT There are two typical deployment scenarios for P2P-VoD services: software and set-top box deployment. In software based systems like PPLive, most users contribute their upload bandwidth to others peers only when they are watching a movie; while in set-box deployment, those set-top boxes are online even when users are not watching any movie. We consider both deployment cases. However, due to page limit, in this section we focus on software deployment; interested readers may refer to our technical report [16] for analysis and design principles in set-top box deployment. We define two operating modes based on the average upload capacity u and the playback rate r: we call the system operates at deficit mode when u<r, or surplus mode when u>r. In addition, we further divide surplus mode into two modes: (a) slightly-surplus mode, i.e., u r, and (b) highly-surplus mode, i.e., u >> r. We like to note that the deficit mode is not a rational operating point for a software P2P-VoD system. If the system operates in deficit mode, the server s total upload bandwidth consumption is at least N(r u), so that the system cannot be scalable with huge number of viewers. The only way to reduce the workload in S is to lower the playback rate r so that the system can be in the surplus mode. On the other hand, if the system operates in the highly-surplus mode, concurrent peers can support each other with ease, so that the abundance of peers upload bandwidth makes replication unnecessary. This abundance in resource will motivate the P2P-VoD system designer to increase the playback rate r so as to improve video quality,which eventually brings the system to the slightlysurplus mode. To summarize, the only interesting case for the software-based P2P-VoD deployment is when the system is operating at the slightly-surplus mode. As we will show, designing an optimal replication strategy is a non-trivial task. Under the software deployment scenario, by setting α =1 in Eq. (9), we have the following proposition: Proposition 2: In software deployment, the average upload bandwidth consumption at the server S is: U(q) = nk =N q k N! ρn1 1 K n 1!...ρnK n K! K l=1 K Dk (n k ) k=1 (C u l (n l )+D l (n l ) n l r) +. (11) To minimize U, the decisions on q k are highly dependent on the shape and characteristics of D k (n k ). We consider some typical distributions to represent peers upload capacity so as to gain a deeper understanding. In addition, we also use numerical and fitting methods to show the characteristics of D k (n k ). In here, we present one typical example. Example 1 (Exploring the characteristics of D k (n k )): Assume n k peers are watching movie M k, and the movie playback rate is r = 5 kbps. According to [4], we set a realistic upload capacity distribution as illustrated in Tab. II. We plot the deficit workload in Fig. 1 when varying n k [1, 1]. It shows that D k (n k ) is concave in n k and can be fitted into the form D k (n k )=D Ae λ(n k n ). Due to page limit, we omit other cases which we have studied, but we state the following observation: bandwidth (kbps) share 5% 3% 5% 15% TABLE II: Peers upload capacity distribution Observation 1: D k (n k ) is a concave function on movie popularity n k, and it converges to a fixed value when n k approaches infinity. Physical meaning: The concavity and convergence features imply that for unpopular movies, they have a much higher marginal increase on the deficit bandwidth D k (n k ) while for popular movies, the marginal increase on D k (n k ) is minimal. 129

5 deficit bandwidth (Mbps) numerical result fitting result D k (n k )=1.36!.75e!.4(n k!.16).4 number of concurrent viewers Fig. 1: Deficit bandwidth D k vs. n k An intuitive explanation of this feature is that peers within a larger set can cooperate more effectively and thus achieve higher utilization of peers upload bandwidth. Using law of large number on Eq. (7), for large enough i, we have i 1 j=1 u j (i 1)u with high probability. Furthermore, if we assume u>rand i>r/(u r)+1, then we have i 1 d c i min (i 1)u d c j,r min ((i 1)(u r),r)=r, j=1 which means that any late arriving peer can receive sufficient download from concurrent peers and will not incur any bandwidth consumption on S. The above analysis shows the reason why deficit bandwidth D k (n k ) is sub-linear. This also implies that the conventional wisdom of using the proportional replication strategy, or caching the more popular movies [5], [17], [18], is suboptimal. Looking at Fig. 1, we see that a popular movie with 25 viewers needs nearly the same deficit bandwidth as a movie with 1 viewers, while an unpopular movie of five viewers needs more than half of the deficit bandwidth as compared with a popular movie with 1 viewers. Now the impact of movie popularities on server s workload is clear: without replication, the server s workload for movie M k is simply D k (n k ). The goal of replication is to reduce the workload on S. The non-linearity of D k (n k ) implies one should be greedy in replicating unpopular movies. In the following, we use examples to justify our argument, and at the same time, determine the optimal replication ratios q. Example 2 (Sub-optimality of the proportional replication strategy): Assume that the system consists of N = 6 active peers and K = 25 movies. Movie M 1 - M 5 are popular movies while movie M 6 - M 25 are unpopular. Peers choose each of the popular movies with a probability of ρ pop =1/6, or each of the unpopular movies with a probability of ρ unpop =1/12. The playback rate is set at r = 5 kbps, Peers upload capacity follows Tab. II. Denote q unpop as the fraction of cache space (which can store 6 movies) that is dedicated to cache any unpopular movie (M 6 - M 25 ) while 1 q unpop is the fraction of cache space of caching popular movies (M 1 - M 5 ). We develop a simulator to explore the replication strategy. In particular, we vary the fraction of cache space that is used to store unpopular movies from % (i.e., all cache space is used to proportional replication optimal replication fraction of caching unpopular movies Fig. 2: Replication impact on server s workload in a softwaredeployed system, where r = 5 kbps. store popular movies) to 1% (i.e., all cache space is used to store unpopular movies), and evaluate the system performance. Fig. 2 illustrates the upload bandwidth consumption on the server when we vary q unpop. If the system only replicates popular movies, then all deficit bandwidth D k of unpopular movies will contribute to the server s workload. This corresponds to the first point from the left (about 15 Mbps) in Fig. 2. Similarly, if all replications are for unpopular ones, the server s workload is around 6 Mbps and this corresponds to the last point at the right of the figure. Since 5/6 fraction of the requests are for popular movies, using the proportional replication strategy, only 1/6 or 17% of the cache space should be dedicated to store the unpopular movies. From the figure, we see that the workload on the server will be around 1 Mbps. However, the optimal replication fraction for unpopular movie should be around 7% where the server s upload bandwidth consumption is less than 2 Mbps. This simple example validates the following observation: Observation 2: In a software-based system, proportional replication is far from optimal and incurs a higher server s workload as compared with the optimal replication strategy. Example 3 (Exploring optimal replication ratio): We consider a P2P-VoD system with 7 peers and 35 movies. M 1 - M 5 are popular movies with ρ pop =1/7; M 6 - M 15 are moderately popular movies with ρ mid =1/7; while M 16 - M 35 are unpopular movies with ρ unpop =1/14. We simulate this system and explore the server s workload at each possible integral fraction combination q {(q pop,q mid,q unpop ) : q pop + q mid + q unpop = 1%}. Using exhaustive search, the popularity fractions and optimal replication fractions of different movies are shown in Tab. III. Obviously, the optimal replication strategy should be aggressive in caching unpopular movies. popular moderate unpopular proportional fraction 71% 14% 14% optimal replication fraction 18% 32% 5% deficit bandwidth fraction (ˆq k ) 21% 3% 49% TABLE III: Optimal replication ratios vs. popularity fractions It is mathematically difficult to derive the optimal ratios of replication from the optimization problem, and one cannot afford to use exhaustive search for large scale P2P-VoD sys- 121

6 tems. However, we obtain an interesting finding: the optimal ratio for replication for movie M k should be close to the fraction of deficit bandwidth D k (n k ) for movie M k among the total deficit bandwidth, i.e., let ˆq k = D k (n k ) K i=1 Di(ni), then we have q opt k ˆq k. For instance, in the above example ˆq pop = 21%, ˆq mid = 3% and ˆq unpop = 49%, which are near to the optimal ratios for replication. This example reveals the following observation: Observation 3: Proportional deficit bandwidth replication is a good estimate to the optimal replication ratios. Due to page limit, we omit examples using other typical settings, but they all strongly support our observations. We like to point out that there is an intuitive explanation to the above resource allocation strategy: without replication, D k (n k ) is exactly the server s bandwidth consumption for movie M k, thus, allocating remaining bandwidth in this ratio is the most efficient method to balance each movie s request load to the server S. This idea leads to the replication algorithm which we will discuss in the next section. IV. ALGORITHMS TO CONTROL REPLICATION RATIOS Let us present several algorithms to achieve the optimal replication ratio q in a P2P-VoD system. Note that a P2P-VoD is with high churn, therefore, the replication algorithms need to be robust and efficient in minimizing the server s workload. In general, there are two ways to control or adjust the replication ratios: Passive adjustment: peers adjust the replication ratio in the P2P-VoD system using replacement algorithms, i.e., when a peer s local cache is full, it decides which movie(s) to purge so as to accommodate a new movie that can reduce the server s workload; Active adjustment: the server S and all peers actively push out (or upload) data for which the replicas are not at the desired replication ratios. Passive Adjustment via Replacement Algorithm: When the cache space of a peer is full, a peer has to decide which movie(s) to purge from the cache space such that new movies could be cached. Since we want to explore the impact of movie popularities to the server s workload, we consider replication algorithms that replace the whole movie, instead of partial replacement. Nevertheless, generalization can be easily made. Previous work [5] suggested a weight-based evaluation scheme, or replicating proportionally to a movie s popularity. Our previous examples already show the sub-optimality of such replication algorithm and one should replicate movie proportionally to its deficit bandwidth. This leads to the following replacement algorithm, which is illustrated in Algorithm 1. The main idea behind this algorithm is to keep the number of replicas proportional to the deficit bandwidth. We define satisfaction index (SI) to express the extent that a movie has enough replicas. When SI k < 1, it means that the number of replicas for movie M k is below the desired replication ratio. On the other hand, if SI k > 1, the number of replicas of M k exceeds the desired ratio. Therefore, in each round, we purge the movie that has the highest SI from the local cache. Algorithm 1 Replacement Algorithm 1: for any peer that starts to watch a new movie do 2: if this movie is already stored in its local cache then 3: do not replace any movie; 4: else 5: for each movie M i in its local cache do 6: calculate the expected number of caches for M i : N exp i =[D i (n i )N] / k D k(n k ); 7: calculate the current number of caches for M i : Ni cur ; 8: calculate the satisfaction index for M i : SI i = Ni cur /N exp i ; 9: end for 1: replace the movie with the highest satisfaction index. 11: end if 12: end for Algorithm 2 Push Algorithm 1: while system is in an idle time slot do 2: find out a list of replication peer candidates which will most likely be inactive at the publishing instance; 3: estimate the total volume available for push 4: calculate the total volume desired for coming-to-hot movies: k D k(n k ) (movie length) 5: if total available volume is larger than desired then 6: assign each movie M i with the volume it desires; 7: else 8: for each movie M i do 9: assign volume D i(n i) k D k(n k ) (total available volume) to the list of replication peer candidates; 1: end for 11: end if 12: end while We like to mention that although requiring global information, this algorithm is efficient in practice. Each peer reports to the tracker the current cached movies, and the tracker can calculate the satisfaction indices and broadcast to all peers. These can be done in a proper frequency (e.g., once in five minutes), and thus the overhead can be minimal. In Sec. V, we also evaluate the impact of delay in broadcasting SI. Active Adjustment via Push Strategy: Although our proposed replacement algorithm can reduce the server s workload, other factors still constrain the system performance. For instance, the server s workload may dramatically increase when a popular movie is newly released, since there will be a sudden surge of demand but only few peers in the system have cached this movie. By simply relying on the replacement algorithm, the system will take a long time to converge to the optimal replication ratios. To overcome this limitation, one can take a proactive approach to push out a movie if we know that it will be a popular object. This leads to the push algorithm as 1211

7 replica distribution replica distribution movie ID 5 4 (a) proportional distribution x 1! movie ID (b) optimal distribution Fig. 3: Proportional vs. optimal replica distribution depicted in Algorithm 2. In essence, the push algorithm utilizes the excessive bandwidth in idle time slots to push some data for future use so server s workload could be flattened over time. Note that there is a daily periodicity of user behavior [5], leading it possible to predict the peak hours and popularity dynamics (in particular, for TV series) so as to implement the algorithm in practice. We will illustrate the effect of the algorithm in the following section. V. PERFORMANCE EVALUATION A. Performance of Replacement Algorithm We carry out extensive simulation to validate our models and evaluate the performance of our proposed replacement algorithm. For a software-based P2P-VoD system where α = 1, we use the following as inputs to our simulator: (a) The system contains N = 1, active peers. Peer s upload capacity distribution follows Tab. II, with average of 531 kbps. (b) The system provides K = 25 kinds of movies. Each movie is with the playback rate of r = 5 kbps. (c) Movie popularity follows a Zipf distribution [9] with parameter γ. (d) At each round, an active peer requests a movie. Active peers switch movie channel every round. The movie playback rate and average upload capacity we set are based on realistic settings like PPLive. The tracker in the system records the current cache distribution in the system and that all peers can use this information to do replacement. We compare server s workload using the following replacement algorithms: (a) Our proposed algorithm, which keeps replicas proportional to deficit bandwidth; (b) Proportional algorithm, which keeps the replicas proportional to movie popularity; (c) First-in-First-out (FIFO) algorithm, which keeps the newly cached movie and purges the oldest one. Fig. 3 illustrates the comparison of the proportional replica distribution vs. the optimal replica distribution (both normalized to 1). We set the Zipf distribution parameter γ = 1 and arrange the movies in a decreasing order of popularity (a) γ =1 (b) γ =1.5 Fig. 4: Comparison of replacement algorithms fraction of satisfied peers Fig. 5: Impact of replication algorithm on satisfied peers Our proposed algorithm keeps fewer popular movies than the proportional distribution and is more greedy in replicating unpopular movies. Thus, the optimal replica distribution is flatter than the proportional distribution. Fig. 4 compares the server s workload using different replacement algorithms. We record the server s load after round 2 since we are interested in the steady state performance. In Fig. 4(a) where γ =1, the server s average workload is around 9 Mbps using our proposed replacement algorithm, but is 75 Mbps using the or 74 Mbps using the. Hence, our proposed push algorithm reduces the server s workload by a factor of eight. When movies popularities have high variability, the performance of our replacement algorithm shows even higher workload reduction. Fig. 4(b) illustrates this result with γ = 1.5, or higher variance in movies popularity. In the steady state, the server s average workload is around 1, 13 and 11 Mbps using our proposed replacement algorithm, proportional algorithm and, respectively. Another important performance measure is to evaluate the number of peers which can watch movie at the normal playback rate r (which we call satisfied peers) when given a finite server upload capacity. We set the simulation environment as the previous case and the server upload capacity is restricted at 1212

8 , p=.5, p=.7 (a) Comparison of algorithms with random delay 6 4 2!=1!=2!=3 (c) Improved algorithm with multiple cache spaces (b) Comparison of algorithms with incomplete information Fig. 6: Practical issues (d) Comparison of algorithms with peer departure 1 Mbps. Fig. 5 shows the comparison on fraction of satisfied peers using different replacement algorithms. From the figure, our proposed replacement algorithm can ensure the maximum ratio of peers that can watch the movie at the playback rate, achieving approximately 1% of satisfied peers, while the ratio is around 96% when using the proportional or. These results validate that our proposed replacement algorithm can reduce the server s workload and at the same time, improve streaming quality. Practical issues: let us comment on some practical implementational issues and the adaptiveness of our algorithms. First, we consider the possible delay when a peer receives information from the tracker. We assume that a peer has probability p to successfully update SI when doing replacement. If this is delayed, a peer will do replacement based on SI it received previously. In Fig. 6(a), we show that our algorithm is robust to this delay: when p =.7 or.5, the server s workload is much less than proportional or s. Secondly, we consider the case that some peers fail to report its cached movie to the tracker, and thus the replacement is based on incomplete information. Fig. 6(b) shows the comparison of server s workload with 1% failure to report. The server s workload is 16 Mbps when using our proposed algorithm, but will be 8 Mbps or 77 Mbps with proportional or. Compared with Fig. 4(a), our proposed algorithm is still robust with incomplete information. Thirdly, when there are more cache space, the server s workload can be further reduced. In Fig. 6(c), we allow each peer to store up to three movies it has watched. The server s average workload is near zero when the variance of movie popularity is not large (e.g., γ =1or 2), and is only around 4 Mbps even if the movie popularities vary a lot (e.g., γ =3). Lastly, peers may not watch the entire movie but may leave after watching part of the movie. This behavior can lead it a more challenging task in supporting latter parts of movies. Theoretically one can use chunk level replication, but it comes with an overhead. In here, we do a minor modification on our proposed algorithm, i.e., divide a movie into several segments (we use three as default) and separately replicate these segments based on individual deficit workload. In Fig. 6(d), we assume that among all peers that watch the first part of any movie M k, 8% of them also watch the second segment, and only 5% watch the third segment. It shows that our proposed algorithm is adaptive to this system, while the proportional or performs poorly. B. Performance of Push Algorithm In here, we show that the proposed push algorithm can significantly improve the system performance, particularly for a set-top box deployed system. The key idea is to push some data to some inactive peers in advance such that the server s workload could be flattened down when publishing a new popular movie. The simulation setting is similar to the previous one but the differences are: (a) There are 5, active peers and 5, inactive peers. (b) Playback rate of a movie is doubled to r =1. Mbps. (c) Initially there are K = 249 kinds of movies {M 1,..,M 249 } with a Zipf popularity distribution (γ = 1). At the beginning of round t p, a new popular movie M 25 is published. The popularities of these 25 movies follow a new Zipf distribution and M 25 is the most popular one. Movie M 1 - M 249 can be stored in local cache of active and inactive peers. In a traditional system, before round t p, movie M 25 will not be in any local cache since no peer requests this unpublished movie. Hence, the system will have a very heavy workload at the publishing instance. With the push strategy, the new movie M 25 can be pushed into inactive peers cache before being published. We divide a day into 24 rounds (1 hour/round), of which we allow 18 rounds in advance to push a new movie before the publication. 1213

9 server workload (Mbps) w/o push strategy w/ push strategy in a day Fig. 7: Performance improvement of using the push strategy To avoid introducing much overhead, in each round we only push the data of this new movie into forty inactive peers. At t p = 19, the new movie is published. After the publishing instance of the new movie, we do not push the data any more. Fig. 7 illustrates the improvement using the push strategy. With a limited overhead, this strategy can significantly decrease the server s workload at the publishing instance of a new movie. In other words, this allows the system to prepare the data ahead of the flash crowd event. In comparison, the system without using the push strategy will experience heavy workload not only at the time of the new movie publication, but also in the following rounds after publishing the movie, as it is slow to adjust to the optimal replication ratio. VI. RELATED WORK Replication has long been an effective strategy to improve performance in P2P systems. Existing literatures [3], [6], [7], [12] discussed replication replacement strategies to optimize file availability, minimize traveling distance or maximize search effectiveness. They vary in objectives and hence lead to different conclusions. Recent popularity of P2P- VoD systems leads to the rethinking of replication. Previous replication algorithms are not suitable due to a very different objective: to minimize the server s workload. Authors in [4] showed the benefit of a peer-assisted VoD system by utilizing peers upload bandwidth, but peers only redistribute content to concurrent peers and hence there is no real replication. Huang et al. [5] discussed the design and implementation of a P2P-VoD system and showed replication can reduce server s workload; but the proportional replication policy was reported to have poor performance for unpopular movies in [13]. Similar proportional replications were also implemented in [17], [18]. Poon et al. [8] reformulated the replication problem in terms of minimizing server s workload. Cheng et al. [2] proposed and evaluated a heuristic algorithm of lazy replication. These two works were based on simulations but did not provide theoretical analysis. Tan et al. [11] discussed content placement for a P2P-VoD system. Wu et al. [15] used dynamic programming to derive the optimal replacement strategy for P2P-VoD system. These two works assumed homogenous upload capacity, which is different from our model which considers peer heterogeneity. Besides, our model differentiates concurrent peers and replication peers which is different from [15]. Suh et al. [1] considered push-to-peer scheme for a set of always-online homogeneous peers, where the content server only pushes out data and the peers only pull contents. This differs from our scheme where we let heterogeneous peers to push out data proactively. Chen et al. [1] considered pre-fetching strategy for a peer-assisted IPTV system based on a structured IPTV platform, which is different from unstructured P2P systems. VII. CONCLUSION Movie replication is important in the design of P2P-VoD systems. However, it remains an open problem to answer what the optimal replication policy should be so as to minimize the server s workload. In this paper, we present mathematical models and formulate an optimization framework to understand the impact of movies popularities on server s workload, and reveal important principles in designing optimal replication algorithms. We show that conventional proportional replication strategy is far from optimal; rather, one should be more aggressive to replicate unpopular movies. Furthermore, to operate the system at the optimal point, we propose both passive replacement and active push strategies. We discuss practical implementational issues and validate via extensive simulations to show that we achieve high QoS guarantee in streaming and at the same time, reduce workload at the server. REFERENCES [1] Y.-F. Chen, Y. Huang, R. Jana, H. Jiang, M. Rabinovich, J. Rahe, B. Wei, and Z. Xiao. Towards capacity and profit optimization of video-ondemand services in a peer-assisted iptv platform. Multimedia System, 15(1):19 32, 29. [2] B. Cheng, L. Stein, H. Jin, and Z. Zhang. A framework for lazy replication in p2p vod. In Proc. of NOSSDAV, 28. [3] E. Cohen and S. Shenker. Replication strategies in unstructured peerto-peer networks. In Proc. of ACM SIGCOMM, 22. [4] C. Huang, J. Li, and K. W. Ross. Can internet video-on-demand be profitable? In Proc. of ACM SIGCOMM, 27. [5] Y. Huang, T. Z. Fu, D.-M. Chiu, J. C. S. Lui, and C. Huang. Challenges, design and analysis of a large-scale P2P-VoD system. In Proc. of ACM SIGCOMM, 28. [6] J. Kangasharju, K. Ross, and D. Turner. 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